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Finite element error estimates for subsonic flow

Published online by Cambridge University Press:  17 February 2009

S. S. Chow  and
G. F. Carey
S. S. Chow
Affiliation:
Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, Texas, U.S.A..
G. F. Carey
Affiliation:
Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, Texas, U.S.A..
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Abstract

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Error estimates are derived for a finite element analysis of plane steady subsonic flows described by the full potential equation. The analysis is based on the use of the theory of variational inequalities to accomodate the subsonic flow constraint and leads to a suboptimal estimate relative to that obtained for linear potential flow. We then consider an alternative dual formulation of the problem and obtain an optimal estimate subject to reasonable regularity assumptions.

Type
Research Article
Information
The ANZIAM Journal , Volume 29 , Issue 1 , July 1987 , pp. 88 - 102
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Bers, L., Mathematical aspects of subsonic and transonic gas dynamics (Interscience, New York, 1958).Google Scholar
[2]Brezis, H. and Stanipacchia, G., “The hodograph method in fluid mechanics in the light of variational inequalities”, Arch. Rational Mech. Anal. 61, 1 (1976), 1–18.CrossRefGoogle Scholar
[3]Bristeau, M., Glowinski, R., Periaux, J., Perrier, P., Pirfonneau, O., and Poirier, G., “Transonic flow simulation by finite element and least square methods”, in Finite elements in fluids Vol. 4, Gallagher, R. H. et al. , (eds.) (Wiley, New York, 1982).Google Scholar
[4]Carey, G. F. and Pan, T. T., “Computation of subcritical compressible flows”, Computers and Fluids 9 (1981), 33–41.CrossRefGoogle Scholar
[5]Carey, G. F. and Pan, T. T., “Shock free redesign using finite elements”, Comm. Appl. Num. Meth. (in press) (1985).Google Scholar
[6]Dinh, H. and Carey, G. F., “Approximate analysis of regularized compressible flow”, J. Nonlin. Anal. (in review) (1984).Google Scholar
[7]Dinh, H. and Carey, G. F., “Some results concerning approximation of regularized compressible flow”, Internal. J. Numer. Methods Fluids 5 (1985), 299–302.CrossRefGoogle Scholar
[8]Falk, R. S., “Approximate solutions of some variational inequalitis with order of convergence estimates”, Ph.D. Thesis, Cornell University, Ithaca, New York, 1971.Google Scholar
[9]Falk, R. S. and Mercier, B., “Error estimates for elasto-plastic problems”, RAIRO Anal. Numer. 11, 2 (1977), 135–144.CrossRefGoogle Scholar
[10]Finn, R. and Gilbarg, D., “Asymptotic behavior and uniqueness of plane subsonic flows”, Comm. Pure Appl. Math. 10 (1957), 23–63.CrossRefGoogle Scholar
[11]Germain, P., “Ecoulements transsoniques homogènes”, 5 Prog. in aeronautical sci. (1964), 143–273.CrossRefGoogle Scholar
[12]Glowinski, R., Lions, J. L. and Tremoliers, R., “Numerical analysis of variational inequalities”, (North Holland, Amsterdam, 1982).Google Scholar
[13]Guderley, K. G., The theory of transonic flow (Addison-Wesley, 1962) (English Translation).Google Scholar
[14]Hafez, M., Weilford, C., Murman, E., “Mixed finite element methods and dual iterative methods for transonic flow”, in Finite elements and fluids, Vol. 3, Gallagher, R. H. et al. , (eds.) (Wiley, New York, 1978).Google Scholar
[15]Li, K.-T. and Huang, A. X., “Solvability of the partial differential equation satisfied by the stream function in compressible flow and the error bound in the finite element solution”, in Finite element flow analysis, Kawai, Tadahiko (ed.) (North-Holland, Amsterdam, 1982) 387–394.Google Scholar
[16]Morawitz, C. S., “On the non-existence of continuous transonic flows past profiles”, Parts I, II Comm. Pure Appl. Math. 10 1957, 45–68.Google Scholar
[17]Shapiro, A. H., The dynamics and thermodynamics of compressible fluid flow, Vols. 1, 2. (Ronald Press, New York, 1953).Google Scholar
[18]Shen, S. F., “Transonic aerodynamics”, in Finite elements and fluids, Vol. 3, Gallagher, R. H. et al. , (eds.) (Wiley, New York, 1978).Google Scholar
[19]von Mises, R., Mathematical theory of compressible fluid flow (Academic Press, New York, 1958).Google Scholar